GHZ state, spin squeezed state, and spin coherent state for frequency estimation under general Gaussian noises

Abstract Exploring the quantum advantages of various non-classical quantum states in noisy environments is a central subject in quantum sensing. Here we provide a complete picture for the frequency estimation precision of three important states (the Greenberger–Horne–Zeilinger (GHZ) state, the maximal spin squeezed state, and the spin coherent state) of a spin- S under both individual dephasing and collective dephasing by general Gaussian noise, ranging from the Markovian limit to the extreme non-Markovian limit. Whether or not the noise is Markovian, the spin coherent state is always worse than the classical scheme under collective dephasing although it is equivalent to the classical scheme under individual dephasing. Moreover, the maximal spin squeezed state always give the best sensing precision (and outperforms the widely studied GHZ state) in all cases. This establishes the general advantage of the spin squeezed state for noisy frequency estimation in many quantum sensing platforms.